2tan (problem 3.3.2)

Average Error: 37.1 → 0.6

Time: 14.3s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

\[\tan \left(x + \varepsilon\right) - \tan x\]

↓

\[\left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)\right) + \tan x \cdot 0\]

C

double f(double x, double eps) {
        double r162076 = x;
        double r162077 = eps;
        double r162078 = r162076 + r162077;
        double r162079 = tan(r162078);
        double r162080 = tan(r162076);
        double r162081 = r162079 - r162080;
        return r162081;
}

↓

double f(double x, double eps) {
        double r162082 = x;
        double r162083 = sin(r162082);
        double r162084 = eps;
        double r162085 = cos(r162084);
        double r162086 = 2.0;
        double r162087 = pow(r162085, r162086);
        double r162088 = r162083 / r162087;
        double r162089 = sin(r162084);
        double r162090 = pow(r162089, r162086);
        double r162091 = 1.0;
        double r162092 = pow(r162083, r162086);
        double r162093 = r162092 * r162090;
        double r162094 = cos(r162082);
        double r162095 = pow(r162094, r162086);
        double r162096 = r162095 * r162087;
        double r162097 = r162093 / r162096;
        double r162098 = r162091 - r162097;
        double r162099 = r162098 * r162094;
        double r162100 = r162090 / r162099;
        double r162101 = r162092 / r162095;
        double r162102 = r162101 + r162091;
        double r162103 = r162098 * r162085;
        double r162104 = r162089 / r162103;
        double r162105 = r162102 * r162104;
        double r162106 = fma(r162088, r162100, r162105);
        double r162107 = r162083 / r162099;
        double r162108 = r162083 / r162094;
        double r162109 = r162107 - r162108;
        double r162110 = r162106 + r162109;
        double r162111 = tan(r162082);
        double r162112 = 0.0;
        double r162113 = r162111 * r162112;
        double r162114 = r162110 + r162113;
        return r162114;
}

Error

Bits error versus x

Bits error versus eps

Target

Original	37.1
Target	15.1
Herbie	0.6

\[\frac{\sin \varepsilon}{\cos x \cdot \cos \left(x + \varepsilon\right)}\]

Derivation

1. Initial program 37.1

   \[\tan \left(x + \varepsilon\right) - \tan x\]
2. Using strategy rm

3. Applied tan-sum 21.8

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon}} - \tan x\]
4. Using strategy rm

5. Applied add-cube-cbrt 22.3

   \[\leadsto \frac{\tan x + \tan \varepsilon}{1 - \tan x \cdot \tan \varepsilon} - \color{blue}{\left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}}\]

6. Applied flip-- 22.3

   \[\leadsto \frac{\tan x + \tan \varepsilon}{\color{blue}{\frac{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}{1 + \tan x \cdot \tan \varepsilon}}} - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\]

7. Applied associate-/r/ 22.3

   \[\leadsto \color{blue}{\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)} \cdot \left(1 + \tan x \cdot \tan \varepsilon\right)} - \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right) \cdot \sqrt[3]{\tan x}\]

8. Applied prod-diff 22.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 \cdot 1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}, 1 + \tan x \cdot \tan \varepsilon, -\sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right)}\]

9. Simplified 22.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}, \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right), -\tan x\right)} + \mathsf{fma}\left(-\sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}, \sqrt[3]{\tan x} \cdot \left(\sqrt[3]{\tan x} \cdot \sqrt[3]{\tan x}\right)\right)\]

10. Simplified 21.8

    \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}, \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right), -\tan x\right) + \color{blue}{\tan x \cdot 0}\]
11. Using strategy rm

12. Applied tan-quot 21.8

    \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\color{blue}{\frac{\sin x}{\cos x}} \cdot \tan \varepsilon\right)}, \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right), -\tan x\right) + \tan x \cdot 0\]

13. Applied associate-*l/ 21.8

    \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \left(\tan x \cdot \tan \varepsilon\right) \cdot \color{blue}{\frac{\sin x \cdot \tan \varepsilon}{\cos x}}}, \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right), -\tan x\right) + \tan x \cdot 0\]

14. Applied associate-*r/ 21.8

    \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \color{blue}{\frac{\left(\tan x \cdot \tan \varepsilon\right) \cdot \left(\sin x \cdot \tan \varepsilon\right)}{\cos x}}}, \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right), -\tan x\right) + \tan x \cdot 0\]

15. Simplified 21.8

    \[\leadsto \mathsf{fma}\left(\frac{\tan x + \tan \varepsilon}{1 - \frac{\color{blue}{\left(\sin x \cdot \tan \varepsilon\right) \cdot \left(\tan x \cdot \tan \varepsilon\right)}}{\cos x}}, \mathsf{fma}\left(\tan \varepsilon, \tan x, 1\right), -\tan x\right) + \tan x \cdot 0\]

16. Taylor expanded around inf 22.0

    \[\leadsto \color{blue}{\left(\left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} + \left(\frac{\sin x \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos \varepsilon\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x\right)} + \left(\frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon} + \frac{{\left(\sin x\right)}^{2} \cdot \sin \varepsilon}{{\left(\cos x\right)}^{2} \cdot \left(\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon\right)}\right)\right)\right) - \frac{\sin x}{\cos x}\right)} + \tan x \cdot 0\]

17. Simplified 0.6

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)\right)} + \tan x \cdot 0\]

18. Final simplification 0.6

    \[\leadsto \left(\mathsf{fma}\left(\frac{\sin x}{{\left(\cos \varepsilon\right)}^{2}}, \frac{{\left(\sin \varepsilon\right)}^{2}}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x}, \left(\frac{{\left(\sin x\right)}^{2}}{{\left(\cos x\right)}^{2}} + 1\right) \cdot \frac{\sin \varepsilon}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos \varepsilon}\right) + \left(\frac{\sin x}{\left(1 - \frac{{\left(\sin x\right)}^{2} \cdot {\left(\sin \varepsilon\right)}^{2}}{{\left(\cos x\right)}^{2} \cdot {\left(\cos \varepsilon\right)}^{2}}\right) \cdot \cos x} - \frac{\sin x}{\cos x}\right)\right) + \tan x \cdot 0\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x eps)
  :name "2tan (problem 3.3.2)"
  :precision binary64

  :herbie-target
  (/ (sin eps) (* (cos x) (cos (+ x eps))))

  (- (tan (+ x eps)) (tan x)))